Two theorems on graceful trees

Let T be a tree on n vertices which are labelled by the integers in N = {1,2,…,n} such that each vertex of T is associated with a distinct number in N. The weight of an edge is defined to be the absolute value of the difference between the two numbers labelled at its end vertices. If the weights of all edges of T are distinct, we call T a graceful tree. In this note, two methods for constructing bigger graceful trees from a given one and a given pair of graceful trees are provided.     

Methionine synthase reductase polymorphisms are associated with serum osteocalcin levels in postmenopausal women

Homocysteine (Hcy) is thought to play an important role in the development of osteoporosis and fracture. Methionine synthase reductase (MTRR) is an enzyme involved in the conversion of Hcy to methionine. We hypothesized that certain genetic polymorphisms of MTRR leading to reduced enzyme activity may cause hyperhomocysteinemia and affect bone metabolism. We therefore examined the associations of the A66G and C524T polymorphisms of the MTRR gene with bone mineral density (BMD) and serum osteocalcin levels in postmenopausal women. Although we did not detect any significant associations between MTRR polymorphisms and BMD or serum osteocalcin levels, we found that the 66G/524C haplotype, which has reduced enzyme activity, was significantly associated with serum osteocalcin levels in a gene-dose dependent manner (P = 0.002). That is, the highest osteocalcin levels (34.5 +/- 16.8 ng/ml) were observed in subjects bearing two copies, intermediate osteocalcin levels (32.6 +/- 14.4 ng/ml) were observed in subjects bearing one copy, and the lowest levels of osteocalcin (28.8 +/- 10.9 ng/ml) were observed in subjects bearing no copies. These results suggest that the 66G/524C haplotype of the MTRR gene affect bone turn over rate.     

Eta1 and eta2 complexes of lambda3-1,2,4,6-thiatriazinyls with CpCr(CO)x

The title heterocyclic radicals coordinate to either 17e CpCr(CO)3 or 15e CpCr(CO)2 moieties as one-electron or as three-electron donors, respectively; in the former the bonding is via the perpendicular p orbital of the sulfur atom, while in the latter bonding is via p(pi) orbitals on both sulfur and nitrogen.     

Euler calculations with embedded Cartesian grids and small-perturbation boundary conditions

This study examines the use of stationary Cartesian mesh for steady and unsteady flow computations. The surface boundary conditions are imposed by reflected points. A cloud of nodes in the vicinity of the surface is used to get a weighted average of the flow properties via a gridless least-squares technique. If the displacement of the moving surface from the original position is typically small, a small-perturbation boundary condition method can be used. To ensure computational efficiency, multigrid solution is made via a framework of embedded grids for local grid refinement. Computations of airfoil wing and wing-body test cases show the practical usefulness of the embedded Cartesian grids with the small-perturbation boundary conditions approach.     

Extension theorems for paraboloids in the finite field setting

In this paper we study the L ^p ? L ^r boundedness of the extension operators associated with paraboloids in ${{\mathbb F}_{q}^{d}}In this paper we study the L ^p − L ^r boundedness of the extension operators associated with paraboloids in \mathbb F_q^d{{\mathbb F}_{q}^{d}} , where \mathbbF_q{\mathbb{F}_{q}} is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples (x,y,z,w) ? E⁴{(x,y,z,w) \in E^4} with x + y = z+w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from L ^p to L ⁴ in the case when −1 is a square number in \mathbbF_q{\mathbb{F}_{q}} . Using the sharp L ^p −L ⁴ result, we improve upon the range of exponents r, for which the L ² − L ^r estimate holds, obtained by Mockenhaupt and Tao (Duke Math 121:35–74, 2004) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in \mathbbF_q{\mathbb{F}_{q}}, we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.     

Magnetic iron oxide nanoparticles for biorecognition: evaluation of surface coverage and activity

Modifying the surfaces of magnetic nanoparticles (MNPs) by the covalent attachment of biomolecules will enable their implementation as contrast agents for magnetic resonance imaging or as media for magnetically assisted bioseparations. In this paper we report both the surface coverage and the activity of IgG antibodies on MNPs. The antibodies were immobilized on gamma-Fe2O3 nanoparticles by conventional methods using aminopropyltriethoxy silane and subsequent activation by glutaraldehyde. Novel fluorescence methods were used to provide a quantitative evaluation of this well-known approach. Our results show that surface coverage can be stoichiometrically adjusted with saturated surface coverage occurring at approximately 36% of the theoretical limit. The saturated surface coverage corresponds to 34 antibody molecules bound to an average-sized MNP (32 nm diameter). We also show that the immobilized antibodies retain approximately 50% of their binding capacity at surface-saturated levels.     

Triode field emitters with planar carbon-nanoparticle cathodes

We designed and fabricated three different types of triode field emitters with planar carbon-nanoparticle (CNP) cathodes such as a normal-gate structure, a double-gate structure, and a well-structure. A normal-gate structure CNP triode emitter showed good field emission properties. The field emission started at the gate-voltage of 45 V, and the anode current reached the level of 120 nA at the gate-voltage of 60 V. However, in general, normal-gate structure suffered from large gate current. Using the double-gate structure, we successfully reduced the gate current to the level less than 4% of anode currents up to the anode current of 250 nA. To simplify fabrication process while maintaining the gate current reduction effect of the double-gate structure, a triode emitter with a well-structure cathode was fabricated via reactive-ion etching of a heavily doped n-type silicon wafer. The triode emitter with a well-structure cathode and a recessed gate structure showed negligible gate current.